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Conditions for growth and extinction in matrix models with environmental stochasticity

Luis Sanz

Abstract

In this kind of model, the main characteristic that determines population viability in the long term is the stochastic growth rate (SGR) denoted $λ_S$. When $λ_S$ is larger than one, the population grows exponentially with probability one and when it is smaller than one, the population goes extinct with probability one. However, even in very simple situations it is not possible to calculate the SGR analytically. The literature offers some approximations for the case in which environmental variability is low, and there are also some lower and upper bounds, but there is no study of the practical situations in which they would be tight. Some new bounds for the SGR are built and the conditions under which each bound works best are analyzed. These bounds are used to give some necessary and some sufficient conditions for population explosion and extinction that are easy to check in practice. The general results are applied to several cases, amongst them a population structured as juveniles and adults living in an environment switching randomly between "rich" and "poor".

Conditions for growth and extinction in matrix models with environmental stochasticity

Abstract

In this kind of model, the main characteristic that determines population viability in the long term is the stochastic growth rate (SGR) denoted . When is larger than one, the population grows exponentially with probability one and when it is smaller than one, the population goes extinct with probability one. However, even in very simple situations it is not possible to calculate the SGR analytically. The literature offers some approximations for the case in which environmental variability is low, and there are also some lower and upper bounds, but there is no study of the practical situations in which they would be tight. Some new bounds for the SGR are built and the conditions under which each bound works best are analyzed. These bounds are used to give some necessary and some sufficient conditions for population explosion and extinction that are easy to check in practice. The general results are applied to several cases, amongst them a population structured as juveniles and adults living in an environment switching randomly between "rich" and "poor".
Paper Structure (13 sections, 1 theorem, 60 equations, 2 figures, 2 tables)

This paper contains 13 sections, 1 theorem, 60 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Matrix models with environmental stochasticity
  3. A particular case: stochastic Leslie model with two age clases
  4. Known bounds for the SGR
  5. A perturbation approach to obtaining bounds for the SGR
  6. Bounds in two-class Leslie models
  7. Bounds on age structure for two-class Leslie models
  8. Bounds for the SGR using info on age structure
  9. Analysis and comparison of the different bounds
  10. A study of growth/extinction in a stochastic Leslie model
  11. Discussion and conclusion
  12. Matrix models with environmental stochasticity
  13. Proofs

Key Result

Theorem 1

Let us consider system (e1) and assume that: a. The set $\{\boldsymbol{A}_{\eta},\ \eta\in\mathcal{I}\}$ of vital rates matrices is ergodic, i.e., there exists a positive integer $g$ such that any product of $g$ matrices (with repetitions allowed) drawn from $\mathcal{A}$ is a positive matrix (i.e., where $\max(\boldsymbol{A}_{\eta})$ is the maximum of the elements of $\boldsymbol{A}_{\eta}$ and $

Figures (2)

  • Figure 1: Mean and standard deviation (SD) of relative error (%) for the best upper bound (UB) and the best lower bound (LB) of $\lambda_{S}$ as functions of relative environmental variation (%)
  • Figure 2: Values of $\log\lambda_{S}$ and of the lower bounds as a function of $\Delta$ in three different configurations.

Theorems & Definitions (1)

  • Theorem 1